What is the weighted normal modifier for?
Left without - right with Weighted Normal Modifier

Let's first explain what Weighted Normal Modifier is doing:


  Face weighted normals (FWN) is a technique to improve shading for 3d
  models. Shading can be improved by adding bevels and altering the
  vertex normals to be perpendicular on the larger flat polygons. This
  forces the shading to be blended across the smaller bevel faces,
  rather than across the entire model.
  http://wiki.polycount.com/wiki/Face_weighted_normals


To visualise this let's look at our example:



You can see here how vertex normals were calculated and gave flat surfaces and nice smooth bevels. Most important thing here is that Bevel has only 1 segment.

Where to use this technique?

Most commonly it is used for low/medium poly assets - games. 

Read here about how it was used in Star Citizen: https://polycount.com/discussion/155894/decal-technique-from-star-citizen

I'm personally heavy using it for arch design. Where you can have whole building with bevels, not so good topography and still results are nice and smooth.

Using it with high poly assets sometimes also could be beneficial especially when topology is really messed up.


  Weighted vertex normals improve the appearance of virtually all
  geometry, and is generally superior to the traditional non-weighted
  average. It works because the undesired shading artifacts are
  displaced from large (highly visible) polygons to their smaller
  neighbors (less visible), and thereby guarantees improved visuals in
  virtually all common situations.
  https://www.bytehazard.com/articles/vertnorm.html


Check also this question with comments: Is it possible to get automatic smooth edges with flat faces without adding geometry?

Bevel modifier:

You can also utilize this method by using Harden Normals in Bevel modifier - AutoSmooth need to be turned on also.